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11 Electrostatics
Fig. 11.44
11.58 Let the charges be q 1 = q 2 on the two spheres before contact.
Force F =
q 1 q 2
4πε 0 r 2 =
q 2
1
4πε 0 r 2
(1)
When the spheres are brought into contact and separated they are at a common potential. Let the new charges be q
1 and q
2 , respectively
q
1 + q
2 = q 1 + q 2 = 2q 1
(from charge conservation)
(2)
Also V =
q
1
4πε 0 R 1
=
q
2
4πε 0 R 2
(3)
or
q
1
q
2
=
R 1
R 2
(4)
Solving (2) and (4)
q
1 =
2q 1 R 1
R 1 + R 2
, q
2 =
2q 1 R 2
R 1 + R 2
(5)
After separation, force F
=
q
1 q
2
4πε 0 r 2
(6)
F
F =
q 2
1
q
1 q
2
=
(r 1 + r 2 ) 2
4r 1 r 2
=
(1 + 3) 2
4 × 1 × 3
=
4
3
(7)
11.59 V =
q
4πε 0 r
∴ q = 4 πε 0 r V =
(5)(2 × 10 4 )
9 × 10 9 = 1.11 × 10
−5 C
11.60 (a) V =
q
4πε 0 a
∴ Capacitance C =
q
V
= 4πε 0 a